Fungrim entry: f61927

T_{n}\!\left(x y\right) \le T_{n}\!\left(x\right) T_{n}\!\left(y\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \left[1, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[1, \infty\right)

References:

V. V. Prasolov, Polynomials, Springer, 2004, ISBN 978-3-642-03980-5. Theorem 3.14.11.
R. Askey, An inequality for Tchebycheff polynomials and extensions, Journal of Approximation Theory, 1975, Volume 14, pp 1-11

TeX:

T_{n}\!\left(x y\right) \le T_{n}\!\left(x\right) T_{n}\!\left(y\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \left[1, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[1, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ZZ`	$\mathbb{Z}$	Integers
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("f61927"),
    Formula(LessEqual(ChebyshevT(n, Mul(x, y)), Mul(ChebyshevT(n, x), ChebyshevT(n, y)))),
    Variables(n, x, y),
    Assumptions(And(Element(n, ZZ), Element(x, ClosedOpenInterval(1, Infinity)), Element(y, ClosedOpenInterval(1, Infinity)))),
    References("V. V. Prasolov, Polynomials, Springer, 2004, ISBN 978-3-642-03980-5. Theorem 3.14.11.", "R. Askey, An inequality for Tchebycheff polynomials and extensions, Journal of Approximation Theory, 1975, Volume 14, pp 1-11"))

Topics using this entry

Chebyshev polynomials